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In this paper we present some classes of topological quantum codes on surfaces with genus $g \geq 2$ derived from hyperbolic tessellations with a specific property. We find classes of codes with distance $d = 3$ and encoding rates asymptotically going to 1, $\frac{1}{2}$ and $\frac{1}{3}$, depending on the considered tessellation. Furthermore, these codes are associated with embedding of complete bipartite graphs. We also analyze the parameters of these codes, mainly its distance, in addition to show a class of codes with distance 4. We also present a class of codes achieving the quantum Singleton bound, possibly the only one existing under this construction.

Current work presents a new approach to quantum color codes on compact surfaces with genus g\geq2 using the identification of these surfaces with hyperbolic polygons and hyperbolic tessellations. We show that this method may give rise to color codes with a very good parameters and we present tables with several examples of these codes whose parameters had not been shown before. We also present a family of codes with minimum distance d=4 and the encoding rate asymptotically going to 1 while n\rightarrow\infty.

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